Statistical Parameters
Two e-sports players have the same mean score per game, but one scores about the same every time and the other alternates between amazing games and total disasters. Which one would you pick for your team? A single number doesn't tell the whole story: you need to know where the center of the data is and how spread out it is.
Mean, median, and mode
These are the measures of central tendency: each one summarizes with a single number where the "center" of the data is.
Change the data and see where the mean (pink line) and the median (green line) end up:
In a study on social media use, a group of teenagers was asked how many hours a day they spend watching videos on TikTok. The results obtained are:…
In a study on social media use, a group of teenagers was asked how many hours a day they spend watching videos on TikTok. The results obtained are:
$2, 3, 1, 4, 2, 3, 2, 5, 2, 1$
Identify the population of the study and calculate the mode of the hours of use.
Seven friends compare the number of unread messages they have on their messaging app. The data are as follows: $12, 15, 8, 20, 15, 14, 11$ Calculate the arithmetic mean…
Seven friends compare the number of unread messages they have on their messaging app. The data are as follows:
$12, 15, 8, 20, 15, 14, 11$
Calculate the arithmetic mean ($\bar{x}$) of the number of unread messages for this group.
During a training week, an athlete recorded the kilometers she ran each day to prepare for a race: $5, 8, 10, 12, 15, 18, 21$ Find the median ($Me$) of the distance…
During a training week, an athlete recorded the kilometers she ran each day to prepare for a race:
$5, 8, 10, 12, 15, 18, 21$
Find the median ($Me$) of the distance covered. What does this value mean in relation to the total number of days trained?
A teacher organised the number of spelling mistakes made by their students in the last dictation in a frequency table: | Mistakes ($x_i$) | No. of students ($f_i$) | |…
A teacher organised the number of spelling mistakes made by their students in the last dictation in a frequency table:
| Mistakes ($x_i$) | No. of students ($f_i$) |
|---|---|
| $0$ | $8$ |
| $1$ | $12$ |
| $2$ | $6$ |
| $3$ | $4$ |
Calculate the mean number of mistakes in the class using the data from the table.
Lucía's marks in her first four Mathematics exams were $6.5$, $4$, $8$ and $7.5$. If she still has a fifth exam to take, what mark must she get for her final mean mark…
Lucía's marks in her first four Mathematics exams were $6.5$, $4$, $8$ and $7.5$. If she still has a fifth exam to take, what mark must she get for her final mean mark to be exactly $7$? Justify your answer by setting up an equation.
Range and mean deviation
Two groups can have the same mean and still be very different: in one, all the data values are close to the mean, and in the other, they are scattered all over the place. Measures of spread capture exactly that: how far apart the data values are from each other or from the mean.
The bars $|\ |$ are the absolute value: we take each distance without its sign, because being 2 below the mean is just as far away as being 2 above it.
Two esports teams recorded their scores in the last 5 matches. Team A: $85, 88, 90, 92, 95$ Team B: $60, 75, 90, 105, 120$ Both teams have the same mean, $\bar{x} = 90$.…
Two esports teams recorded their scores in the last 5 matches.
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Team A: $85, 88, 90, 92, 95$
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Team B: $60, 75, 90, 105, 120$
Both teams have the same mean, $\bar{x} = 90$. Without performing complex standard deviation calculations, look at the data and explain which of the two teams is more consistent (has less dispersion) and why.
A youth clothing shop wants to analyse the age of its customers. To do this, it takes a sample of 10 people on a Saturday morning:…
A youth clothing shop wants to analyse the age of its customers. To do this, it takes a sample of 10 people on a Saturday morning:
$14, 15, 16, 14, 22, 15, 17, 14, 35, 16$
- a) Calculate the range and the mean age.
- b) The median is $15.5$ years. If we remove the 35-year-old customer (which is considered an outlier), calculate the new mean and the new median. Which parameter has changed the most? Why do you think this happens?
Chivas scored these goals in their last $6$ Liga MX games: $2, \ 0, \ 3, \ 1, \ 4, \ 2$ The mean is $2$ goals per game. Find the mean absolute deviation,…
Chivas scored these goals in their last $6$ Liga MX games:
$2, \ 0, \ 3, \ 1, \ 4, \ 2$
The mean is $2$ goals per game. Find the mean absolute deviation, $MAD = \dfrac{\sum |x_i - \bar{x}|}{N}$.
At a middle school in Oaxaca, $10$ students were asked how many books they read over vacation. These are the results: | Books read ($x_i$) | $0$ | $1$ | $2$ | $3$ | |…
At a middle school in Oaxaca, $10$ students were asked how many books they read over vacation. These are the results:
| Books read ($x_i$) | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|
| Students ($f_i$) | $2$ | $3$ | $3$ | $2$ |
Find the mean, the range, and the mean absolute deviation of the number of books read.
Comparing two data sets
To decide between two options using data, looking at one number isn't enough. Calculate a measure of central tendency and a measure of spread for each set, put them side by side, and ask yourself what you need: the highest value, or the most consistent one?
Look at the following two data sets, which represent the points scored by two basketball players in their last 5 games: Player A: $18, 20, 22, 19, 21$ Player B:…
Look at the following two data sets, which represent the points scored by two basketball players in their last 5 games:
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Player A: $18, 20, 22, 19, 21$
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Player B: $10, 30, 5, 25, 30$
In the figure you have the same data represented on a number line (each dot is a game).
Without doing complex variance calculations, look at the data and answer:
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Do they have the same arithmetic mean?
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Which of the two players is more "consistent" in their scoring?
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If you had to choose one of them to play the last minute of a final, which one would you choose and why? Use the concepts of mean and dispersion in your answer.
These are Ximena's and Emiliano's grades on the five math tests of the term: | Student | Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | | :--- | :---: | :---: | :---: |…
These are Ximena's and Emiliano's grades on the five math tests of the term:
| Student | Test 1 | Test 2 | Test 3 | Test 4 | Test 5 |
|---|---|---|---|---|---|
| Ximena | $8$ | $9$ | $8$ | $7$ | $8$ |
| Emiliano | $10$ | $6$ | $9$ | $7$ | $8$ |
- a) Find each student's mean.
- b) Find each student's range and mean absolute deviation.
- c) Which of the two is more consistent? Justify your answer with your results.
Diego lives in Mexico City and can get to school by Metro or by Metrobús. For six days he wrote down how many minutes each option took: Metro:…
Diego lives in Mexico City and can get to school by Metro or by Metrobús. For six days he wrote down how many minutes each option took:
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Metro: $40, \ 42, \ 38, \ 41, \ 39, \ 40$
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Metrobús: $30, \ 46, \ 34, \ 50, \ 32, \ 48$
- a) Find the mean and the mean absolute deviation for each option.
- b) Diego leaves home at $6{:}15$ and school starts at $7{:}00$, with no grace period. Which option should he take? Justify your decision using the mean and the spread of the data.
The Pumas coach has to choose a forward for the final. These are the goals two players scored in the last $8$ matchdays: Player A: $1, \ 1, \ 2, \ 1, \ 0, \ 1, \ 2, \ 0$…
The Pumas coach has to choose a forward for the final. These are the goals two players scored in the last $8$ matchdays:
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Player A: $1, \ 1, \ 2, \ 1, \ 0, \ 1, \ 2, \ 0$
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Player B: $0, \ 3, \ 0, \ 0, \ 4, \ 0, \ 1, \ 0$
- a) Find the mean, the range, and the mean absolute deviation for each player.
- b) In what percentage of the matchdays did each player score at least one goal?
- c) Both players have the same mean. Which one should the coach choose if he wants a player he can count on in every game? Justify your answer using the spread and your result from part b).
Quartiles and box plots
Just as the median splits the ordered data into two halves, the quartiles mark positions around 25%, 50%, and 75%:
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$Q_1$: first quartile, the 25% position.
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$Q_2$: the median (50%).
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$Q_3$: third quartile, the 75% position.
The box plot summarizes everything in one drawing: the "whiskers" go from the minimum to the maximum, and the box goes from $Q_1$ to $Q_3$, with a line at the median. It represents the middle 50%. With few data values or repeated values, these percentages should not be interpreted as exact counts of values that are strictly smaller.
Calculating quartiles step by step
We'll use this convention: put the data in order, find the median, and take $Q_1$ and $Q_3$ to be the medians of the lower and upper halves. If there is an odd number of data values, leave out the median when forming the halves. There are other conventions: state the one you chose and don't mix results from different methods.
The height of a group of 9th grade (3° de secundaria) students was measured and the data were ordered from smallest to largest. The values of the quartiles obtained are:…
The height of a group of 9th grade (3° de secundaria) students was measured and the data were ordered from smallest to largest. The values of the quartiles obtained are:
$Q_1 = 158 \text{ cm}, Q_2 = 165 \text{ cm}, Q_3 = 172 \text{ cm}$
- a) What is the value of the median?
- b) What percentage of students is $172 \text{ cm}$ tall or less?
Observe the following representation of a data set on a number line. The dots represent the number of pets owned by several neighbors: From the graph, write down the…
Observe the following representation of a data set on a number line. The dots represent the number of pets owned by several neighbors:
From the graph, write down the list of data and calculate the mode ($Mo$) and the range.
Variance and standard deviation
Parameters from frequencies
With frequencies, $s^2 = \dfrac{\sum f_i(x_i-\bar{x})^2}{N}$. Don't calculate the mean or the variance of the column of values without taking into account how many times each one appears.
Comparing and deciding
Two teams take, in minutes, A: 8, 10, 12 and B: 2, 10, 18. Both have a mean of 10, but their standard deviations are approximately 1.63 and 6.53. If you need predictable times, A is more consistent. The CV is approximately 0.163 (16.3%) for A and 0.653 (65.3%) for B. Lower spread doesn't mean one team is better for every goal: explain what you need to compare.
Two messaging apps, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) was measured over 10 different attempts, obtaining the…
Two messaging apps, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) was measured over 10 different attempts, obtaining the following results:
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App A: $\bar{x} = 120$ ms; $\sigma = 15$ ms
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App B: $\bar{x} = 125$ ms; $\sigma = 5$ ms
Calculate the coefficient of variation ($CV$) for both applications and explain with reasons which of the two is more "reliable" or consistent in its performance for a user.
In a PE class, the recovery time of two groups after a run was measured. Both groups have the same mean heart rate, $\bar{x} = 85$ bpm. However, the variance of Group 1…
In a PE class, the recovery time of two groups after a run was measured. Both groups have the same mean heart rate, $\bar{x} = 85$ bpm. However, the variance of Group 1 is $S^2 = 16$ and that of Group 2 is $S^2 = 100$.
- a) Calculate the standard deviation of each group.
- b) In which of the two groups do the students have a more similar fitness level to each other? Reason your answer based on the dispersion.